4 color theorum
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  Election What-ifs? (Moderator: Dereich)
  4 color theorum
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MaC
Milk_and_cereal
Junior Chimp
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« on: April 18, 2005, 10:59:12 PM »
« edited: April 18, 2005, 11:00:53 PM by Milk_and_cereal »



With what candidates could there be an electoral map of the four color therom? 
Is there actually a way to make the United States with three colors?  I know it can be done to a certain extent, but I missed out by coloring Indiana, Pennsylvania, and Oregon a different color.  Make me a map, of it if you think you can.
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jfern
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« Reply #1 on: April 18, 2005, 11:02:56 PM »

It's impossible to it with 3 colors. Any state that touches an odd number of other states in a normal way (the four corners area is a special case), and isn't at an edge of the country, for example Nevada, requires 3 colors other than its own for its neighbors.
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MaC
Milk_and_cereal
Junior Chimp
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« Reply #2 on: April 18, 2005, 11:03:19 PM »

note to self, use a dictionary, learn how to spell theorem.
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BRTD
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« Reply #3 on: April 18, 2005, 11:21:16 PM »

some researchers at the University of Illinois-Champaign, actually wrote a proof for that 4 colors is all you need to make a map with no two of the same colors touching, however the proof is more than 600 pages long and a good 60 pages of that is proofs of other proofs they used in the big proof, and would no doubt be boring as hell to analyze.
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jfern
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« Reply #4 on: April 18, 2005, 11:24:40 PM »

some researchers at the University of Illinois-Champaign, actually wrote a proof for that 4 colors is all you need to make a map with no two of the same colors touching, however the proof is more than 600 pages long and a good 60 pages of that is proofs of other proofs they used in the big proof, and would no doubt be boring as hell to analyze.

The original proof was really really ugly. Someone else came up with a merely reall ugly proof.

http://www.math.gatech.edu/~thomas/FC/fourcolor.html#References
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muon2
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« Reply #5 on: April 18, 2005, 11:30:10 PM »

some researchers at the University of Illinois-Champaign, actually wrote a proof for that 4 colors is all you need to make a map with no two of the same colors touching, however the proof is more than 600 pages long and a good 60 pages of that is proofs of other proofs they used in the big proof, and would no doubt be boring as hell to analyze.
One of the great achievements of late 20th century mathematics. The 1977 proof showed that there were important theorems that required a computer to solve. The simper versions of the proof since then have also required a computer in some capacity.
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Erc
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« Reply #6 on: April 19, 2005, 12:16:42 AM »

Proof that three colors is not adequate for the map of the United States:

Let us have our three colors be Red, Blue, and Green (entirely arbitrary).

Kentucky must be one of these colors, and Tennessee must be another.  (Say, Red and Blue respectively).

Missouri and Virginia, as they touch both Red Kentucky and Blue Tennessee, must both be Green.

West Virginia must be Blue, as it touches Red Kentucky and Green Virginia.

Illinois must be Blue, as it touches Red Kentucky and Green Missouri.

Ohio must be Green, as it touches Red Kentucky and Blue West Virginia.

Which leaves us with Indiana, which, if three colors are adequate, must touch three states, each of which have three different colors (Red Kentucky, Green Ohio, and Blue Illinois).  Can't make this state anything but a fourth color.


I have just disproved the three-color theorem.

As other posters have said, proving the four-color theorem is really really hard (although you yourself has proven it applies to this specific case).
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